\name{chronopl}
\alias{chronopl}
\title{Molecular Dating With Penalized Likelihood}
\usage{
chronopl(phy, lambda, age.min = 1, age.max = NULL,
         node = "root", S = 1, tol = 1e-8,
         CV = FALSE, eval.max = 500, iter.max = 500, ...)
}
\arguments{
  \item{phy}{an object of class \code{"phylo"}.}
  \item{lambda}{value of the smoothing parameter.}
  \item{age.min}{numeric values specifying the fixed node ages (if
    \code{age.max = NULL}) or the youngest bound of the nodes known to
    be within an interval.}
  \item{age.max}{numeric values specifying the oldest bound of the nodes
    known to be within an interval.}
  \item{node}{the numbers of the nodes whose ages are given by
    \code{age.min}; \code{"root"} is a short-cut for the root.}
  \item{S}{the number of sites in the sequences; leave the default if
    branch lengths are in mean number of substitutions.}
  \item{tol}{the value below which branch lengths are considered
    effectively zero.}
  \item{CV}{whether to perform cross-validation.}
  \item{eval.max}{the maximal number of evaluations of the penalized
    likelihood function.}
  \item{iter.max}{the maximal number of iterations of the optimization
    algorithm.}
  \item{\dots}{further arguments passed to control \code{nlminb}.}
}
\description{
  This function estimates the node ages of a tree using a
  semi-parametric method based on penalized likelihood (Sanderson
  2002). The branch lengths of the input tree are interpreted as mean
  numbers of substitutions (i.e., per site).
}
\details{
  The idea of this method is to use a trade-off between a parametric
  formulation where each branch has its own rate, and a nonparametric
  term where changes in rates are minimized between contiguous
  branches. A smoothing parameter (lambda) controls this trade-off. If
  lambda = 0, then the parametric component dominates and rates vary as
  much as possible among branches, whereas for increasing values of
  lambda, the variation are smoother to tend to a clock-like model (same
  rate for all branches).

  \code{lambda} must be given. The known ages are given in
  \code{age.min}, and the correponding node numbers in \code{node}.
  These two arguments must obviously be of the same length. By default,
  an age of 1 is assumed for the root, and the ages of the other nodes
  are estimated.

  If \code{age.max = NULL} (the default), it is assumed that
  \code{age.min} gives exactly known ages. Otherwise, \code{age.max} and
  \code{age.min} must be of the same length and give the intervals for
  each node. Some node may be known exactly while the others are
  known within some bounds: the values will be identical in both
  arguments for the former (e.g., \code{age.min = c(10, 5), age.max =
    c(10, 6), node = c(15, 18)} means that the age of node 15 is 10
  units of time, and the age of node 18 is between 5 and 6).

  The input tree may have multichotomies. If some internal branches are
  of zero-length, they are collapsed (with a warning), and the returned
  tree will have less nodes than the input one. The presence of
  zero-lengthed terminal branches of results in an error since it makes
  little sense to have zero-rate branches.

  The cross-validation used here is different from the one proposed by
  Sanderson (2002). Here, each tip is dropped successively and the
  analysis is repeated with the reduced tree: the estimated dates for
  the remaining nodes are compared with the estimates from the full
  data. For the \eqn{i}{i}th tip the following is calculated:

  \deqn{\sum_{j=1}^{n-2}{\frac{(t_j - t_j^{-i})^2}{t_j}}}{SUM[j = 1, ..., n-2] (tj - tj[-i])^2/tj},

  where \eqn{t_j}{tj} is the estimated date for the \eqn{j}{j}th node
  with the full phylogeny, \eqn{t_j^{-i}}{tj[-i]} is the estimated date
  for the \eqn{j}{j}th node after removing tip \eqn{i}{i} from the tree,
  and \eqn{n}{n} is the number of tips.

  The present version uses the \code{\link[stats]{nlminb}} to optimise
  the penalized likelihood function: see its help page for details on
  parameters controlling the optimisation procedure.
}
\value{
  an object of class \code{"phylo"} with branch lengths as estimated by
  the function. There are three or four further attributes:

  \item{ploglik}{the maximum penalized log-likelihood.}
  \item{rates}{the estimated rates for each branch.}
  \item{message}{the message returned by \code{nlminb} indicating
    whether the optimisation converged.}
  \item{D2}{the influence of each observation on overall date
    estimates (if \code{CV = TRUE}).}
}
\references{
  Sanderson, M. J. (2002) Estimating absolute rates of molecular
  evolution and divergence times: a penalized likelihood
  approach. \emph{Molecular Biology and Evolution}, \bold{19},
  101--109.
}
\author{Emmanuel Paradis}
\seealso{
  \code{\link{chronoMPL}}
}
\keyword{models}